## The Implicit Function Theorem: History, Theory, and ApplicationsThe implicit function theorem is part of the bedrock of mathematical analysis and geometry. Finding its genesis in eighteenth century studies of real analytic functions and mechanics, the implicit and inverse function theorems have now blossomed into powerful tools in the theories of partial differential equations, differential geometry, and geometric analysis. There are many different forms of the implicit function theorem, including (i) the classical formulation for C^k functions, (ii) formulations in other function spaces, (iii) formulations for non- smooth functions, (iv) formulations for functions with degenerate Jacobian. Particularly powerful implicit function theorems, such as the Nash--Moser theorem, have been developed for specific applications (e.g., the imbedding of Riemannian manifolds). All of these topics, and many more, are treated in the present volume. The history of the implicit function theorem is a lively and complex story, and is intimately bound up with the development of fundamental ideas in analysis and geometry. This entire development, together with mathematical examples and proofs, is recounted for the first time here. It is an exciting tale, and it continues to evolve. "The Implicit Function Theorem" is an accessible and thorough treatment of implicit and inverse function theorems and their applications. It will be of interest to mathematicians, graduate/advanced undergraduate students, and to those who apply mathematics. The book unifies disparate ideas that have played an important role in modern mathematics. It serves to document and place in context a substantial body of mathematical ideas. |

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### Contents

Preface | ix |

Introduction to the Implicit Function Theorem | 1 |

12 An Informal Version of the Implicit Function Theorem | 3 |

13 The Implicit Function Theorem Paradigm | 7 |

History | 13 |

22 Newton | 15 |

23 Lagrange | 20 |

24 Cauchy | 27 |

Variations and Generalizations | 93 |

52 Implicit Function Theorems without Differentiability | 99 |

53 An Inverse Function Theorem for Continuous Mappings | 101 |

54 Some Singular Cases of the Implicit Function Theorem | 107 |

Advanced Implicit Function Theorems | 117 |

62 Hadamards Global Inverse Function Theorem | 121 |

63 The Implicit Function Theorem via the NewtonRaphson Method | 129 |

64 The NashMoser Implicit Function Theorem | 134 |

Basic Ideas | 35 |

32 The Inductive Proof of the Implicit Function Theorem | 36 |

33 The Classical Approach to the Implicit Function Theorem | 41 |

34 The Contraction Mapping Fixed Point Principle | 48 |

35 The Rank Theorem and the Decomposition Theorem | 52 |

36 A Counterexample | 58 |

Applications | 61 |

42 Numerical Homotopy Methods | 65 |

43 Equivalent Definitions of a Smooth Surface | 73 |

44 Smoothness of the Distance Function | 78 |

642 Enunciation of the NashMoser Theorem | 135 |

643 First Step of the Proof of NashMoser | 136 |

644 The Crux of the Matter | 138 |

645 Construction of the Smoothing Operators | 141 |

646 A Useful Corollary | 144 |

Glossary | 145 |

151 | |

161 | |

### Common terms and phrases

algebra apply assume Banach space calculus Cauchy Ck function coefficients compact complex analysis conclude consider continuously differentiable continuously differentiable function contraction mapping contraction mapping fixed convex Corollary curve defined definition denote directional derivative domain eccentric anomaly Euclidean space example fact follows formulated func function F given Hadamard's holds holomorphic function homotopy hyperplane hypothesis imbedding implicit function theorem implies induction inverse function theorem iteration Jacobian determinant Jacobian matrix Krantz Lagrange Lagrange's Lemma Lipschitz locus of points manifold mapping F mapping fixed point method metric space Nash-Moser theorem nearest point Newton polygon nonsingular nonvanishing notation obtain one-to-one open set ordinary differential equations partial derivatives points satisfying polynomial positive reach power series prove rank theorem real analytic result Section sequence signed distance function smooth solution solve Suppose surface tangent topological space unique variables vector Weierstrass preparation theorem zero